API for ergm Terms
Author: Pavel N. Krivitsky

# API for `ergm` Terms

#### Pavel N. Krivitsky

# Summary

This document seeks to be the most up-to-date API documentation for `ergm` terms. Note that it is not intended to be a tutorial as much as a description of what inputs and outputs different parts of the system expect.

Several annotated examples are provided in the off-CRAN template package `ergm.useters`, currently hosted on GitHub at https://github.com/statnet/ergm.userterms .

# Overview

## Types of storage and types of term

The storage API defines two types of storage: private storage, which is attached to the `ModelTerm` structure and is specific to each `ergm` term, and public storage, which is attached to the `Model` and can be accessed by all terms.

A statistic is a familiar `ergm` term like “`edges`” or “`nodefactor`”: it adds at least one sufficient statistic to the model. Every statistic can have private storage, and it can read from public storage, but it cannot write to public storage.

An auxiliary in an `ergm` term, but it is not an ERGM term in the mathematical sense: it adds no statistics to the model and exists only to initialize and maintain public storage to be used by statistics and other auxiliaries. It may not be specified on an `ergm` formula by the end-user but only requested by a statistic or another auxiliary. Note that in the latter case, circular dependencies are not checked.

## Code path

For the purposes of this overview, the following information is relevant, and is elaborated formally later:

- Each term has a a pointer `void *mtp->storage` to private storage
- Each term has a pointer to an array of pointers to public storage `void **mtp->aux_storage`. (The pointer is to the same location for all terms of a model.)
- Each term has some subset of five functions on the `C` side: initializers (`i_`), updaters (`u_`), change stats (`c_`), difference stats (`d_`), finalizers (`f_`), writers (`w_`), and “eXtended” functions (`x_`).
- An `InitErgmTerm.` function’s output list can have an additional element, `auxiliaries`, a one-sided formula.

1. The user passes a formula to the procedure (`ergm`, `summary`, etc.); this formula may only have statistics.
2. `ergm_model()` is called.
   1. It iterates through the formula terms, calling `InitErgmTerm.<NAME>()` functions in turn (or `InitWtErgmTerm.<NAME>()` for valued ERGMs), adding their output to the model term list. Some terms include `auxiliaries` formulas in their list.
   2. `call.ErgmTerm()` when it finds that a term has requested auxiliaries, attaches an attribute `attr(., "aux.slots")` containing an integer vector for the model’s own and/or requested auxiliaries’ positions on the `aux_storage` vector.
   3. It calls `ergm.auxstorage()` with the complete model.
      1. It iterates through initialized model terms, looking at their `auxiliaries` element for a one-sided formula listing their requested auxiliaries.
      2. It iterates through the auxiliary formulas, calling term initialization on them.
      3. It iterates through the `auxiliaries` elements of auxiliary terms and initialises those, etc..
      4. It constructs a list of unique initialized auxilariy terms (that is, when two or more terms had requested an identical auxiliary).
      5. It inserts the initialized unique auxiliary terms in `model$terms` and the index (in the unique list) of the auxiliary requested by each statistic in the `aux.slots` of the requesting statistic.
      6. For an auxiliary itself, the first element of `aux.slots` is set to the position of the auxiliary itself, i.e., the public storage element for which it is responsible.
3. The `ergm_state` is constructed from an edgelist (`state$el`), an empty network (`state$nw0`), a model (`state$model`), and (optionally) a proposal (`state$proposal`) and a statistics vector (`state$stats`).
4. `update.ergm_state()` is called.
   1. It iterates through the formula terms, calling `term$ext.encode()` (if defined) to construct a vector `state$ext.state`. `state$ext.flag` is set to reconciled.
5. The `ergm_state` is passed to the `C` code.
6. `Redgelist2Network()` initializes the network.
7. `ModelInitialize()` is called:
   1. `ModelInitialize()` initializes all terms (statistic and auxiliaries), also counting up the number of auxiliaries (distinguished by having no `c_`, `d_`, or `s_` functions). A term can export both a `c_` function and a `d_` function. In that case, it is responsible for deleting one of them when its `i_` function is called.
   2. It counts up the number of auxiliaries and allocates an array of `void *`s, one for each auxiliary. The `mtp->aux_storage` pointer for each term is set to point to that (one) array.
   3. It assigns `attr(model$terms[[i]], "aux.slots")` to `mtp->aux_slots`.
   4. It assigns the `ergm_model` `SEXP` to `model->R`, in case it’s needed.
   5. It assigns the `ergm_model$terms[[i]]` `SEXP` to `mtp->R`, in case it’s needed.
   6. It calls `InitStats()`, which calls the initializer (`i_` function) of each term or, if not found, an updater (`u_` function) with invalid input (i.e., toggle \((0,0)\)) is called in case the term developer prefers a one-function implementation.
   - The terms are initialized in reverse order, so auxiliaries are initialized before the statistics are, and statistics and auxiliaries can count on their auxiliaries being initialized by the time they are initialized.
8. For each iteration, a proposal is made.
   1. `ChangeStats()` is called.
      1. It calls the `d_` functions, for those terms for which they are initialized.
      2. It iterates through the proposed toggles.
         1. It calls the `c_` functions.
         2. It adds its output to the cumulative change.
         3. It calls the `u_` functions with the toggle (if more to come).
         4. It makes the toggle provisionally (if more to come).
      3. It undoes the provisional toggles.
   2. If the proposal is accepted, `u_` function is called, and network is updated for each toggle.
9. If the term has an `s_` function but neither `c_` nor `d_`, `SummStatsS()` is called after each MCMC interval.
10. `ErgmStateRSave()` is called:
    1. `Network2Redgelist()` is called, returning a `SEXP` with the state.
    2. For each term, a writer (`w_` function) is called if defined, returning a `SEXP` with the extended state.
    3. These are stored in a vector `state$ext.state`. Element `state$ext.flag` is set to signal that a change was made on the `C` side.
11. `ModelDestroy()` is called:
    1. `DestroyStats()` is called, iterating through the terms.
       1. Finalizer (`f_`) function is called, if defined.
       2. If `mtp->storage` is not `NULL`, it is freed.
    2. Other parts of the model are freed; in particular those of `mtp->aux_storage` if not `NULL`.
12. `NetworkDestroy()` is called.
13. States and other outputs are passed back to `R`.
14. `update.ergm_state()` is called.
    1. It iterates through the formula terms, calling `term$ext.decode()` (if defined) to update `state$nw0` or other aspects of the network. `state$ext.flag` is set to reconciled.

# Formal API definition

## Evaluation API

### `R` side

The following is adapted from the header of `R/InitErgmTerm.R`

#### `InitErgmTerm.*` and `InitWtErgmTerm.*` functions

The following are the minimal arguments of these functions:

`function(nw, arglist, ...)`

##### Expects

`nw`: a `network` (or `networkLite`) object taken from the LHS of the model formula or the `basis=` argument (if given); valued networks are instrumented with `%ergmlhs% "response"` information.

`arglist`: a `list` of arguments passed to the term call on the formula. If arguments are passed with names, the list is named. Typically, helper function `check.ErgmTerm()` is used to preprocess this list with the equivalent of R’s `match.call()`. Notably, thanks to R’s lazy evaluation, it is possible to obtain the argument expressions without evaluating them by calling `substitute(arglist)`. See `InitErgmTerm.:` and `InitErgmTerm.*` in `R/InitErgmTerm.interaction.R` for examples.

`env`: an environment, typically of the `formula` from which the term was extracted.

Any term options are passed as direct arguments (not `arglist`). See `options?ergm` for details.

##### Returns

Three return types are possible:

1. `NULL`, to indicate that this term does not add to the model. (E.g., `nodefactor("a", levels=FALSE)`.)
2. An `ergm_model` object, in which case its terms are “pasted” into the model. This is useful for some term operators.
3. A named `list` defining the properties of the term. The following names have special meanings; but any other elements can be included and will be accessible on the `C` side.

In the following, let \(p = \dim(\eta) = \dim(g(y))\), the dimension of the statistic \(g(y)\) being computed and of the canonical parameter \(\eta\). If the model is curved, the model parameter vector \(\theta\) is first mapped onto \(\eta=\eta(\theta)\); otherwise, \(\eta\equiv\theta\). Let \(q=\dim(\theta)\).

`name` (required): a string containing the term’s `C`-side name: `ergm` will search for `name` prepended with `"c_"` for change statistics, `"d_"` for difference, `"s_"` for summary, etc.. This is the only required element.

`coef.names`: a character vector of length \(p\) of names for the elements of the canonical statistic of the model (and canonical parameters); can be absent or zero-length for auxiliaries.

`inputs`: a vector of (double-precision) numeric inputs that will be made available to the `C`-side implementation as a `double` vector; optionally, `inputs` may have an attribute named `"ParamsBeforeCov"`, which is used primarily for backwards compatibility, but can also be used to more conveniently separate, e.g., metadata elements from vertex attribute elements.

`iinputs`: a vector of integer inputs that will be made available to the `C`-side implementation as an `int` vector; optionally, `iinputs` may have an attribute named `"ParamsBeforeCov"`, which is used primarily for backwards compatibility, but can also be used to more conveniently separate, e.g., metadata elements from vertex attribute elements.

`pkgname`: a string containing the name of the `R` package containing the `C` implementation; if not specified, the package in which the `Init*ErgmTerm.*` function was found is assumed.

`dependence`: a logical value indicating whether the addition of this term to the model induces dyadic-dependence; if all terms (ignoring auxiliaries) have `dependence` set to `FALSE`, the model is inferred to be dyad-independent; if not specified, `TRUE` is assumed. A special value of `NA` infers dependence from the auxiliaries used by the term: if all of its auxiliaries are independent, then the term is independent.

`emptynwstats`: a numeric vector of length \(p\) providing the value of the statistic if evaluated on an empty network; if not specified or NULL, assumed to be a vector of zeros. (See `InitErgmTerm.degree()` in `R/InitEergmTerm.R` for an example.)

`minpar` and `maxpar`: numeric vectors of length \(q\) giving the bounds on the valid values for the model’s parameters; if not specified, `-Inf` and `+Inf` vectors are assumed.

`offset`: a logical vector of length \(q\) that allows the term to mark some of its own statistics as having fixed parameters.

For curved terms, all of the following must be present except `cov`:

`params`: a list whose names correspond to element names of the curved parameter vector \(\theta\); the items in the list are there for historical reasons and are ignored.

`map`: a function taking at least two arguments, `x` (a numeric \(q\)-vector containing \(\theta\)), `n`\(=p\) (the length of the output) and an optional `cov` parameter; it is to return a numeric vector of length `n` containing \(\eta(\theta)\).

`gradient`: a function taking the same arguments as `map` and returning the gradient of `map` (\(\eta'(\theta)\)) as a \(q\times p\) numeric matrix.

`cov`: an optional arbitrary data structure that if present is passed to `map` and `gradient`.

If the term has an extended state (such as used by the dynamic simulation of `tergm`), it should also provide the following:

`ext.encode`: a function taking two arguments, `el` (an edge list) and `nw0` (a `network` without edges) and returning an arbitrary `R` data structure encoding the additional information about `nw0`.

`ext.decode`: a function taking three arguments, an encoded extended state (typically created on the `C` side by the `w_` function), `el` (an edge list), and `nw0` (a `network` without edges) and returning a list with two elements: `el` with the updated edge list and `nw0` with the updated network.

### `C` side

#### `ModelTerm` and `WtModelTerm` data structures

The data structure contains information accessible to the `C`-side statistics. Here, `outlist` refers to the list returned by the `Init*ErgmTerm.*` function.

A number of its elements are for internal use and should generally not be considered a part of the API or accessed by the term (with some rare exceptions). These include: `c_func`, `d_func`, `i_func`, `u_func`, `f_func`, `s_func`, `w_func`, `x_func`, `z_func`, `statspos`, `statcache`, `emptynwstats`.

Furthermore, elements `aux_storage` and `aux_slots` should not be accessed directly but only with helper described in the auxiliary storage API below.

The following elements are a part of the API:

`double *attrib` (`INPUT_ATTRIB`, `DINPUT_ATTRIB`): contents of `outlist$inputs`, shifted by `ParamsBeforeCov` attribute if given. (Rarely used.)

`int *iattrib` (`IINPUT_ATTRIB`): contents of `outlist$iinputs`, shifted by `ParamsBeforeCov` attribute if given. (Rarely used.)

`int nstats` (`N_CHANGE_STATS`): value of \(p\), the length of the statistic vector.

`double *dstats` (`CHANGE_STAT`): a \(p\)-vector to be overwritten with statistic value.

`int ninputparams` (`N_INPUT_PARAMS`, `N_DINPUT_PARAMS`): length of `outlist$inputs`.

`double *inputparams`: contents of `outlist$inputs`.

`int niinputparams` (`N_IINPUT_PARAMS`): length of `outlist$iinputs`.

`int *iinputparams` (`IINPUT_PARAMS`): contents of `outlist$iinputs`.

`void *storage` (`STORAGE`): a pointer managed by the term to its private storage space.

`unsigned int n_aux` (`N_AUX`): number of auxiliaries associated with this term; typically the number requested, plus one if the term is itself an auxiliary.

`SEXP R`: the contents of `outlist` as an `R` expression.

`SEXP ext_state`: Location of the extended state information for the term. See below.

#### `c_`, `d_`, and `s_` functions

`d_` functions are the original difference statistics. `c_` functions are new, while `s_` functions have been around for a long time, but never formally documented.

Change statistic (binary): `void c_<NAME>(Vertex tail, Vertex head, ModelTerm *mtp, Network *nwp, Rboolean edgestate)`

Change statistic (valued): `void c_<NAME>(Vertex tail, Vertex head, double weight, WtModelTerm *mtp, WtNetwork *nwp, double edgestate)`

Difference statistic (binary): `void d_<NAME>(Vertex *tails, Vertex *heads, ModelTerm *mtp, Network *nwp)`

Difference statistic (valued): `void d_<NAME>(Vertex *tails, Vertex *heads, double *weights, WtModelTerm *mtp, WtNetwork *nwp)`

Summary statistic (binary): `void s_<NAME>(ModelTerm *mtp, Network *nwp)`

Summary statistic (valued): `void s_<NAME>(WtModelTerm *mtp, WtNetwork *nwp)`

##### Expects

Parameters

Note: In undirected networks, it can be assumed that `tail` < `head` (and similarly with the multiple toggles). In bipartite networks, `tail`s are in the first partition and `head`s are in the second.

`Edge ntgoggles`: Number of edges to be toggled or updated.

`Vertex tail`: Tail of (1) dyad to be toggled or updated.

`Vertex *tails`: An array of tails of the dyads to be toggled or updated.

`Vertex head`: Head of (1) dyad to be toggled or updated.

`Vertex *heads`: An array of heads of the dyads to be toggled or updated.

`double weight`: New weight for (1) dyad.

`double *weights`: An array of new weights for the dyads to be toggled or updated.

`ModelTerm *mtp`: A pointer to the `ModelTerm` data structure. See `inst/include/ergm_changestat.h` and `inst/include/ergm_wtchangestat.h` for details.

`Network *nwp`: A pointer to the `Network` of interest before any toggles are applied.

`Rboolean edgestate`: An indicator of whether edge `(tail,head)` is in the network `nwp` pre-toggle.

`double edgestate`: The weight of dyad `(tail,head)` in the network `nwp` pre-update.

Storage

All functions except for `s_` expect any storage they need to be initialized and up to date (consistent with `nwp`). In particular, if their statistic requested \(k\) auxiliary terms, the \(k\) (`mtp->n_aux`) elements of its `mtp->aux_slots` vector will be the indexes of `mtp->aux_storage` where they can find the respective objects.

##### Macros

It is worth noting that macros defined for `d_` functions that refer to a specific toggle, such as `TAIL`, `HEAD`, etc. might not be usable in a `c_` function, but it’s made up for by `c_` function’s reduced need for bookkeeping: `tail`, `head`, etc. can be used directly.

##### Side-effects

These functions overwrite `mtp->dstats` (often aliased as `CHANGE_STAT`) with the following:

- `c_` and `d_` functions: change of the value of the statistic they implement relative to `nwp` due to the toggles.
- `s_` the value of the statistic it implements.

## Storage API

Every `ergm` term has private storage, found at `void *mtp->storage`, which allows it to store arbitrary information about the state of the network, as well as precalculated values of variables, preallocated memory it needs for its calculations, or any other use. It does so by specifying an updating function (and, optionally, an initialization and a finalization function). This updating function is called every time the network is about to change. The API for these functions is defined below.

Public storage is found at `void **mtp->aux_storage`. Each auxiliary term gets assigned a slot (i.e., `void *nwp->mtp->aux_storage[i]`) to manage; its slot number is the first element of its `aux_slots` vector, and terms requesting it are told which slot to look in in a similar fashion. An auxiliary term that requests other auxiliaries will have its own slot as the first element of `aux_slots` and the slots of auxiliaries it requests as subsequent elements.

### `R` side

A statistic that only references its private storage or is an auxiliary itself does not need to do anything special on the `R` side.

To request an auxiliary, a term’s `InitErgmTerm` call’s output list must include an `auxiliaries` element containing a one-sided `ergm`-style formula listing the auxiliary terms it wishes to use separated by the `+` operator.

### `C` side: Modifying Storage

#### `i_` functions: Initializer/Constructor

This function is optional for using storage: if it’s not provided, the model code will call the `u_` function with an invalid toggle first, signaling for it to initialize. If the term makes use of the extended state API, it is also typically up to the `i_` function to decode the term’s extended state created by `term$ext.encode()` into a useful `C` structure in its private or public storage.

Binary: `void i_<NAME>(Model *mtp, Network *nwp)`

Valued: `void i_<NAME>(WtModel *mtp, WtNetwork *nwp)`

##### Expects

In general, `i_` function expects to be called after `ModelInitialize()` and `NetworkInitialize()`, before any `c_` or `d_` functions. That is, the network must be populated with the ties of its initial state and have `mtp->aux_storage` vector allocated.

Private storage

Network populated with initial ties and initialized model.

Public storage

For an auxiliary, the first element of `mtp->aux_slots` is the index of the element of `mtp->aux_storage` that it is to manage. That is `mtp->aux_storage[mtp->aux_slots[0]]` is a `void *` to point to the data to be public.

##### Side-effects

Private storage

Allocates memory for the information to be stored and overwrites `mtp->storage` with a pointer to it, then updates the stored information to be consistent with `*nwp`.

Public storage

An auxiliary allocates memory for the information to be stored and overwrites `mtp->aux_storage[mtp->aux_slots[0]]` with a pointer to it, then updates the stored information to be consistent with `*nwp`.

An auxiliary can also use its private storage as needed.

#### `u_` functions: Updater

Binary: `void u_<NAME>(Vertex tail, Vertex head, Model *mtp, Network *nwp, Rboolean edgestate)`

Valued: `void u_<NAME>(Vertex tail, Vertex head, double weight, Model *mtp, Network *nwp, double edgestate)`

##### Expects

Initialized network. If no `i_` function was provided, to be called with a \((0,0)\) toggle as a signal to initialize; otherwise, initialized storage. Any statistic or auxiliary can rely on its auxiliaries having been initialized before it.

##### Side-effects

If called with an a toggle `(0,0)` and uninitialized storage, initialize. This will never be done if an `i_` function is defined for the term.

Update the state of its storage (`mtp->storage` and/or `mtp->aux_storage[mtp->aux_slots[0]]` for an auxiliary) to match what the state of `*nwp` would be after the given dyad had been toggled.

#### `f_` functions: Finalizer/Destructor

This function is optional for using storage: if it’s not provided, the model code will free any pointers to `mtp->aux_storage` and `mtp->storage` that are not `NULL`.

Binary: `void f_<NAME>(Model *mtp, Network *nwp)`

Valued: `void f_<NAME>(WtModel *mtp, WtNetwork *nwp)`

##### Expects

Network and a model.

##### Side-effects

Deallocates its storage (`mtp->storage` and/or `mtp->aux_storage[mtp->aux_slots[0]]`) and sets its pointers to `NULL`.

### `C` side: Accessing Storage

#### Private storage

`c_`, `d_`, and `s_` functions can read from, but not write to, their private storage. `c_` and `d_` functions can rely on initialization having been called before.

#### Public storage

Auxilaries must not implement `c_`, `d_`, and `s_` functions.

Terms requesting one or more auxiliaries will be passed the indices of the element of `mtp->aux_storage` by inserting them at the start of `mtp->aux_slots`. That is `mtp->aux_storage[mtp->aux_slots[0]]` is a `void *` to point to the data public by the first auxiliary term on the `auxiliaries` formula, `mtp->aux_storage[mtp->aux_slots[1]]` is the second, etc..

## `x_` functions: eXtensions

This interface is intended to be used by packages extending `ergm` to send arbitrary signals to statistics and auxiliaries. For example, for temporal ERGMs, it may be used to signal to the statistic that the clock is about to advance. It is the responsibility of the extension writer to ensure that everything behaves sensibly.

Binary: `void x_<NAME>(unsigned int type, void *data, Model *mtp, Network *nwp)`

Valued: `void x_<NAME>(unsigned int type, void *data, Model *mtp, Network *nwp)`

##### Expects

Parameters

`type`: a magic constant identifying the type of signal being sent. Based on it, the function can ignore the signal, or determine how to interpret `data`.

`data`: arbitrary data to be sent to the function; it is up to the extension writer to determine how it is formatted and interpreted.

##### Side-effects

There are no restrictions on side-effects. It is up to the extension writer to ensure that everything works.

## `w_` functions: save extended state

This interface is intended to be used by packages extending `ergm` to send information back other than the edge list back to `R`. For example, for dynamic simulation of temporal ERGMs, it is used to send back information about the last toggle time of each extant edge. It will typically encode the information in its private or public storage and return an `R` object that will be used by `term$ext.decode()` to update the template network `nw0` and edge list `el` of the `ergm_state`.

Binary: `SEXP w_<NAME>(Model *mtp, Network *nwp)`

Valued: `SEXP w_<NAME>(WtModel *mtp, WtNetwork *nwp)`

# Macros

The following helper macros have been defined to date, and can be found in `storage.h`.

## Memory management

These functions are defined in `ergm_storage.h` and exported.

`ALLOC_STORAGE(nmemb, stored_type, store_into)`: Allocate a vector of `nmemb` elements of type `stored_type`, save its pointer to private storage and also to a `stored_type *store_into` which is also declared. Should be used by the `i_` function, but may also be used by the `u_` function.

`GET_STORAGE(stored_type, store_into)`: Declare `stored_type *store_into` and assign the pointer to private storage to it. Can be used by all functions.

`ALLOC_AUX_STORAGE(nmemb, stored_type, store_into)`: Allocate a vector of `nmemb` elements of type `stored_type`, and save it to the auxiliary storage slot belonging to the calling auxiliary and into a `stored_type *store_into` which is also declared. Can be used by the `i_` function, but may also be used by the `u_` function.

`GET_AUX_STORAGE(stored_type, store_into)`: Declare `stored_type *store_into` and assign the pointer to the auxiliary storage (either for a statistic or for the auxiliary). Can bn used by all functions.

`GET_AUX_STORAGE(ind, stored_type, store_into)`: Declare `stored_type *store_into` and assign the pointer to the `ind`th auxiliary). Can be used by clients of auxiliaries.

## Miscellaneous helpers

`ALLOC_AUX_SOCIOMATRIX(stored_type, store_into)`: Allocate an array of appropriate dimension with elements of type `stored_type`, save it to auxiliary storage, and into `**store_type`, so that `store_into[i][j]` returns the value associated with dyad \((i,j)\), with vertices indexed from 1. For bipartite and undirected networks, as little space as possible (resp. a rectangle or a triangle) is allocated.

Note that this term assumes that the private and the public storage of the calling term are not used in any other way.

`FREE_AUX_SOCIOMATRIX`: Frees the sociomatrix allocated by `ALLOC_AUX_SOCIOMATRIX`.

# MHproposal storage API

## Auxiliaries

MHproposals may also request auxiliary terms. An `InitErgmProposal.<NAME>()` or `InitWtErgmProposal.<NAME>()`with an `auxiliaries` formula will similarly receive the positions of its auxiliaries’ slots in the network. However, this appears to have a slight cost in speed and a potentially significant cost in memory, since the auxiliary may need to duplicate the information in the `MH_` function.

## Private storage

Functions prefixed with `Mi_`, `Mu_`, and, `Mf_` serve as respectively the initializers, the updaters, and the finalizers of the MHproposal storage, though the old-style call with `MHp->ntoggles==0` is also supported. Macros in the `ergm_MHstorage.h` header file can be used to access storage the same way as for the statistics.

The function called to generate the proposal can have a prefix of either `MH_` (for backwards compatibility) or `Mp_` for consistency.

One important difference is that `Mp_` function is permitted to write to its private storage. This may be useful if, say, a systematic sample is desired.