Descendant distributions for the impact of mutant contagion on networks

Abstract

Contagion, broadly construed, refers to anything that can spread infectiously from peer to peer. Examples include communicable diseases, rumors, misinformation, ideas, innovations, bank failures, and electrical blackouts. Sometimes, as in the 1918 Spanish flu epidemic, a contagion mutates at some point as it spreads through a network. Here, using a simple susceptible-infected model of contagion, we explore the downstream impact of a single mutation event. Assuming that this mutation occurs at a random node in the contact network, we calculate the distribution of the number of "descendants," d, downstream from the initial "patient zero" mutant. We find that the tail of the distribution decays as d −2 for complete graphs, random graphs, small-world networks, networks with block-like structure, and other infinite-dimensional networks. This prediction agrees with the observed statistics of memes propagating and mutating on Facebook and is expected to hold for other effectively infinite-dimensional networks, such as the global human contact network. In a wider context, our approach suggests a possible starting point for a mesoscopic theory of contagion. Such a theory would focus on the paths traced by a spreading contagion, thereby furnishing an intermediate level of description between that of individual nodes and the total infected population. We anticipate that contagion pathways will hold valuable lessons, given their role as the conduits through which single mutations, innovations, or failures can sweep through a network as a whole.

Are fast test results preferable to high test sensitivity in contact-tracing strategies?

“…To quantify the impact of test-trace-isolate strategies on growing epidemics, we simulate the branching structure of the chains of infections, also referred to as the “epidemic tree”. 18,19 Our model lets us keep track of who infected whom, which is essential when simulating contact tracing and quarantining infectious people. In the model (Figure 1A), infected individuals give rise to new cases, unless quarantined following testing and tracing efforts.…”

Motifs for processes on networks

“…The motifs in these models can be process motifs or structure motifs. Researchers have used DAGs to describe the spread of behavior, norms, and ideas [70] and the spread of infectious diseases [71,72] on networks. One can view subgraphs of these so-called "dissemination trees" [70] and "epidemic trees" [71,72] as process motifs.…”

Motif-based mean-field approximation of interacting particles on clustered networks

“…With applications in as disparate branches of science as statistical physics [1], epidemiology [2][3][4][5][6][7][8][9], chemistry and systems biology [10,11], social science [12,13], and computer science [14][15][16], interacting particles on complex networks constitute an important class of models in the mathematician's and physicist's toolkit [17][18][19][20]. They describe systems where individual entities (particles), endowed with local states, interact with a subset of other entities (neighbors) and transition from one state to another as time evolves.…”