Local Structure and Network Epidemics

Network epidemics refers to the the spreading of a disease across nodes in a network or graph where edges represent contacts between nodes that could transmit infection. This differs from many traditional methods for modeling epidemics such as metapopulation models or traditional mass-action model, all of which make implicit assumptions about the underlying network of connections. The explicit contact network allows for more granular modeling, particularly when modeling ``what-if'' scenarios.

The challenge with more detailed network models is that specifying them is challenging. For compartmental models such as SIS or SIR, the potential state space is exponential in the number of nodes. Thus, much research on network epidemics resorts to mean-field approximations to reduce this to a much smaller deterministic system at the cost of fine-scaled information. Even in cases where direct simulation is used, it can be difficult to reach a mechanistic understanding of how fine-grained network structure impacts spreading and interventions. This primarily lies in the ambiguity in precisely defining notions such as structure in networks without overly specifying a fixed size-resolution.

In order to address this, we provide a tool to summarize local network structure based on the Network Community Profile (NCP) and carry out extensive discrete event simulations across a variety of networks. While doing so, we introduce two random models that can plausibly reproduce the local network structure found in empirical networks. We also perform ablation-like studies where we gradually perturb networks and simulate epidemics. Our summary metric is strongly dependent on local conductance bottlenecks in the network that are intimately tied to overlapping community structures. This then avoids the issue of explicitly picking a size-resolution. In carrying out this study, we also investigate whether local structure due to triangles are enough to explain and capture the epidemic effects related to local structure that we observe. We find that there remain key differences between our notion of NCP-based local structure measure and local structure that involves only triangles.

In another direction, we investigate higher-order epidemics, where spread occurs based on a multiway contact among nodes modeled via a hyperedge. Higher-order epidemics are less understood than their pairwise counterparts, with conflicting reports in the literature about how higher-order interactions impact spread. In order to study these questions, we develop a spatial random graph model for interpolating from purely pairwise interactions to higher-order interactions. The hyperedges of this model are based on spatial clusters of points. We then use this model to better understand and probe notions such as hypergraph clustering coefficients and higher-order epidemics. While we make use of a clustering algorithm, we show that the connected components of our model are invariant to both the clustering algorithm and it's parameters. For higher-order epidemics, we find that the outcomes are highly model dependent and one cannot decouple the topology from the dynamics. In particular, injecting higher-order structure could cause total infections to increase, decrease, or stay relatively the same depending on epidemic parameters.

Apart from epidemics, we revisit our adapted Network Community Profile (NCP) and the problem of sampling a network with a given NCP. Relying on it's intimate connection with overlapping community structure, we give a rewiring scheme based on edge swaps that exactly preserves the conductance of a sequence of potentially overlapping and non-exhaustive sets. We formally show that the state space of such graphs is connected. This enables us to produce a random sample of a graph with a given NCP.